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Properties

Label 650.6.b.m
Level $650$
Weight $6$
Character orbit 650.b
Analytic conductor $104.249$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,6,Mod(599,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.599");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 650.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(104.249482878\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 1203x^{8} + 437691x^{6} + 46376305x^{4} + 1526628600x^{2} + 3636090000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \beta_{5} q^{2} + ( - 2 \beta_{5} + \beta_1) q^{3} - 16 q^{4} + ( - 4 \beta_{2} - 8) q^{6} + (\beta_{7} - 9 \beta_{5} + 3 \beta_1) q^{7} + 64 \beta_{5} q^{8} + (\beta_{3} - 6 \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_{5} q^{2} + ( - 2 \beta_{5} + \beta_1) q^{3} - 16 q^{4} + ( - 4 \beta_{2} - 8) q^{6} + (\beta_{7} - 9 \beta_{5} + 3 \beta_1) q^{7} + 64 \beta_{5} q^{8} + (\beta_{3} - 6 \beta_{2} - 2) q^{9} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + \cdots - 43) q^{11}+ \cdots + (176 \beta_{6} - 77 \beta_{4} + \cdots + 88197) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 160 q^{4} - 72 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 160 q^{4} - 72 q^{6} - 8 q^{9} - 426 q^{11} - 336 q^{14} + 2560 q^{16} + 1134 q^{19} - 7400 q^{21} + 1152 q^{24} - 6760 q^{26} - 1080 q^{29} - 21656 q^{31} + 4808 q^{34} + 128 q^{36} - 3042 q^{39} - 22922 q^{41} + 6816 q^{44} - 5088 q^{46} + 14570 q^{49} + 25718 q^{51} + 42264 q^{54} + 5376 q^{56} + 89372 q^{59} + 37944 q^{61} - 40960 q^{64} + 17816 q^{66} + 65436 q^{69} + 47856 q^{71} + 101664 q^{74} - 18144 q^{76} + 135352 q^{79} + 53258 q^{81} + 118400 q^{84} - 16160 q^{86} + 574462 q^{89} - 14196 q^{91} + 280496 q^{94} - 18432 q^{96} + 880568 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 1203x^{8} + 437691x^{6} + 46376305x^{4} + 1526628600x^{2} + 3636090000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -67\nu^{8} - 78646\nu^{6} - 25680107\nu^{4} - 1544967080\nu^{2} - 5558454000 ) / 1085697600 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -67\nu^{8} - 78646\nu^{6} - 25680107\nu^{4} - 1002118280\nu^{2} + 125268106800 ) / 542848800 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2003\nu^{8} + 2317404\nu^{6} + 774539583\nu^{4} + 60394808510\nu^{2} + 769142190600 ) / 3257092800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4609\nu^{9} + 5342622\nu^{7} + 1780200129\nu^{5} + 136322867140\nu^{3} + 2378155471200\nu ) / 3273378264000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9349\nu^{8} + 10501422\nu^{6} + 3271667709\nu^{4} + 181192232020\nu^{2} - 288643334400 ) / 3257092800 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 55966 \nu^{9} + 44307573 \nu^{7} - 1507143444 \nu^{5} - 5614805893895 \nu^{3} - 401535352719900 \nu ) / 3273378264000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 55897 \nu^{9} - 66997196 \nu^{7} - 24175803317 \nu^{5} - 2452201539290 \nu^{3} - 57227206773400 \nu ) / 363708696000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 227191 \nu^{9} + 262613553 \nu^{7} + 87900677121 \nu^{5} + 7048933808635 \nu^{3} + 137457493305300 \nu ) / 818344566000 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 2\beta_{2} - 241 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} + 8\beta_{8} - 8\beta_{7} + 576\beta_{5} - 477\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{6} + 112\beta_{4} - 573\beta_{3} + 1890\beta_{2} + 114745 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -370\beta_{9} - 4816\beta_{8} + 4696\beta_{7} - 509736\beta_{5} + 250757\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -1616\beta_{6} - 73856\beta_{4} + 305093\beta_{3} - 1421338\beta_{2} - 60382849 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -290982\beta_{9} + 2586840\beta_{8} - 2611080\beta_{7} + 371433072\beta_{5} - 134245789\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4963176\beta_{6} + 43765776\beta_{4} - 161561341\beta_{3} + 973901234\beta_{2} + 32372899657 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 421053406\beta_{9} - 1375059104\beta_{8} + 1449506744\beta_{7} - 249998481720\beta_{5} + 72352942965\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
599.1
− 23.8063i
− 7.92606i
1.60591i
8.98932i
22.1371i
− 22.1371i
− 8.98932i
− 1.60591i
7.92606i
23.8063i
− 4.00000i − 25.8063i −16.0000 0 −103.225 − 115.455i 64.0000i −422.964 0
599.2 − 4.00000i − 9.92606i −16.0000 0 −39.7042 124.276i 64.0000i 144.473 0
599.3 − 4.00000i − 0.394093i −16.0000 0 −1.57637 − 194.075i 64.0000i 242.845 0
599.4 − 4.00000i 6.98932i −16.0000 0 27.9573 76.6003i 64.0000i 194.149 0
599.5 − 4.00000i 20.1371i −16.0000 0 80.5485 66.6546i 64.0000i −162.503 0
599.6 4.00000i − 20.1371i −16.0000 0 80.5485 − 66.6546i − 64.0000i −162.503 0
599.7 4.00000i − 6.98932i −16.0000 0 27.9573 − 76.6003i − 64.0000i 194.149 0
599.8 4.00000i 0.394093i −16.0000 0 −1.57637 194.075i − 64.0000i 242.845 0
599.9 4.00000i 9.92606i −16.0000 0 −39.7042 − 124.276i − 64.0000i 144.473 0
599.10 4.00000i 25.8063i −16.0000 0 −103.225 115.455i − 64.0000i −422.964 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 599.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.6.b.m 10
5.b even 2 1 inner 650.6.b.m 10
5.c odd 4 1 650.6.a.o ✓ 5
5.c odd 4 1 650.6.a.p yes 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.6.a.o ✓ 5 5.c odd 4 1
650.6.a.p yes 5 5.c odd 4 1
650.6.b.m 10 1.a even 1 1 trivial
650.6.b.m 10 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 1219T_{3}^{8} + 432963T_{3}^{6} + 45023617T_{3}^{4} + 1306759936T_{3}^{2} + 201867264 \) acting on \(S_{6}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 201867264 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( (T^{5} + 213 T^{4} + \cdots - 310309345876)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 28561)^{5} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 35\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 57\!\cdots\!32)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 21\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 190703824040808)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 35\!\cdots\!60)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 33\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 36\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 99\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 23\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
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